Why could a thin cosmic string shape a whole gravitational-wave spectrum?

A line of trapped energy

The early universe was hot, and as it cooled some fields may have settled into new states, the way water freezes into ice. If neighbouring regions settled differently, the field could stay stuck in its old, high-energy state along thin lines where those regions meet. Those lines are cosmic strings. String theory offers a second route: fundamental strings stretched to cosmic size. Auclair et al., 2020, Section 1

A string is far thinner than an atom but enormously heavy for its width. Its mass per length is about the square of the energy scale at which it formed, and for the simplest strings the tension, the pull along the string, equals that mass per length times the square of the speed of light. Physicists describe the string with one number: its tension multiplied by Newton's gravitational constant and divided by the fourth power of the speed of light. That number, written $G\mu$, has no units. A string from the scale where the strong, weak and electromagnetic forces might unify would have $G\mu$ near $10^{-6}$.

To get a feel for a much lighter string, take $G\mu=10^{-10}$. That is about $1.3\times10^{17}$ kilograms in every metre: the mass of the Earth packed into a string a little longer than the equator.

Loops break off and ring

Strings in the network move at a large fraction of the speed of light. When two pieces cross, they usually swap partners and reconnect. A string that crosses itself cuts off a closed loop. A loop smaller than the distance light has travelled since the Big Bang no longer feels the expansion much. It oscillates under its own tension like a plucked rubber band in space, and it repeats its motion every time light could travel half its length.

As it oscillates, a loop shakes spacetime and sends out gravitational waves. Its shape makes the signal sharper. Loops typically develop cusps, points that briefly move at the speed of light and beam a short burst of waves in one direction. Loops also carry kinks, sudden bends left over from past crossings, which send bursts too. Auclair et al., 2020, Section 4.1

The waves carry energy away at a steady rate that grows with the square of the tension. So the loop shortens at a steady pace until it disappears. A loop born one tenth as long as the distance light has travelled since the Big Bang, with $G\mu=10^{-10}$, lives about twenty million times longer than the age of the universe at its birth. A loop born when the universe was one second old would last about eight months. One born when the universe was a century old would last two billion years.

Every era adds its own pitch

The network keeps its share of the cosmic energy fixed as the universe expands. It does so by turning string into loops, so new loops form in every era. Each new loop is born at a size tied to the age of the universe at that moment. Early loops were therefore tiny and rang at very high pitch. Later loops were huge and rang slowly.

The expansion then moves each signal down in frequency. It stretches every wave by the same factor that space has grown since the wave left. A wave sent out when the universe was a billionth of its present size arrives a billion times longer.

Add up all the loops that ever lived, and the result is a broad hum rather than a single note: a stochastic gravitational-wave background. In its high-frequency part, made by loops from the era when radiation dominated the universe, every doubling of cosmic time contributes about the same amount. The spectrum there is almost flat, and its height grows with the square root of the tension. At lower frequencies the spectrum rises to a bump. For $G\mu=10^{-10}$, a model calculation puts more than nine tenths of the signal at one cycle per year in waves emitted after matter took over from radiation, about fifty thousand years after the Big Bang.

Why tension moves the whole curve

A lighter string radiates more weakly, so you might expect its background simply to drop. It does drop, but not evenly. Weak radiation also means longer lives. Light loops survive longer and pile up in greater numbers, which partly makes up for their weakness. Their final emission happens later, when the universe is larger, and that pushes the bump to higher frequency.

That is why different detectors are sensitive to different strings. For $G\mu=10^{-10}$, the bump sits in the range of pulsar timing arrays, which watch the clock-like signals of pulsars for years. For $G\mu=10^{-17}$, the model signal at one cycle per year is several hundred billion times weaker, while the flat part still reaches the band of the planned LISA space detector at a few thousandths of a hertz. The authors of the main LISA study expect LISA to be sensitive to tensions down to about $10^{-17}$. Auclair et al., 2020, Section 1 and Figure 8

What pulsar timing has and has not shown

In 2023 pulsar timing arrays reported evidence for a gravitational-wave background at nanohertz frequencies. NANOGrav tested whether cosmic strings could explain it. Ordinary stable strings fitted the data worse than pairs of supermassive black holes: the spectrum they predict in that band is either too weak or too flat. Some variants, such as cosmic superstrings that reconnect less often, fitted well, but the collaboration stressed that these results depend strongly on how the black-hole population is modelled and should not be regarded as evidence for new physics. NANOGrav Collaboration, 2023, Sections 1 and 5.4

A curve that passes through a measured signal is not a discovery. Cosmic strings would leave a recognisable pattern across many detectors at once: a rise at low frequency, a long plateau at high frequency, and repeated sharp bursts from cusps. Finding that pattern, or ruling it out, needs pulsar timing, LISA and ground-based detectors together.

To work through the numbers, start the course with what the tension of a cosmic string measures. For the pulsar-timing results in detail, read the guide to the pulsar-timing and LISA results.